If the remainder on division of`x^(3) - kx^(2) + 13x -21` by `-21` . find the quotient and the value of k. Hence, find the zeros of the cubic polynomial `x^(3) -kx^(2) + 13x`.

Answer

To solve the problem step by step, we need to find the value of \( k \) such that the remainder when dividing the polynomial \( x^3 - kx^2 + 13x - 21 \) by \( -21 \) is equal to \( -21 \). We will also find the zeros of the cubic polynomial \( x^3 - kx^2 + 13x \).

### Step 1: Set up the equation
Given the polynomial \( P(x) = x^3 - kx^2 + 13x - 21 \), we want to find \( k \) such that the remainder when \( P(x) \) is divided by \( -21 \) is \( -21 \).

### Step 2: Use the Remainder Theorem
According to the Remainder Theorem, the remainder of a polynomial \( P(x) \) when divided by \( x - a \) is \( P(a) \). Here, we will set \( a = -21 \):

\[ P(-21) = (-21)^3 - k(-21)^2 + 13(-21) - 21 \]

### Step 3: Calculate \( P(-21) \)
Calculating each term:

1. \( (-21)^3 = -9261 \)
2. \( k(-21)^2 = k \cdot 441 \)
3. \( 13(-21) = -273 \)

Putting it all together:

\[ P(-21) = -9261 - 441k - 273 - 21 \]
\[ P(-21) = -9261 - 441k - 273 - 21 = -9261 - 273 - 21 = -9261 - 294 - 441k \]
\[ P(-21) = -9555 - 441k \]

### Step 4: Set the remainder equal to -21
We set the expression equal to -21:

\[ -9555 - 441k = -21 \]

### Step 5: Solve for \( k \)
Rearranging the equation:

\[ -441k = -21 + 9555 \]
\[ -441k = 9534 \]
\[ k = \frac{9534}{441} = \frac{53}{2} \]

### Step 6: Substitute \( k \) back into the polynomial
Now that we have \( k = \frac{53}{2} \), we substitute it back into the polynomial:

\[ P(x) = x^3 - \frac{53}{2}x^2 + 13x \]

### Step 7: Factor the polynomial to find zeros
To find the zeros, we set \( P(x) = 0 \):

\[ x^3 - \frac{53}{2}x^2 + 13x = 0 \]

Factoring out \( x \):

\[ x(x^2 - \frac{53}{2}x + 13) = 0 \]

### Step 8: Solve the quadratic equation
Now we need to solve the quadratic equation:

\[ x^2 - \frac{53}{2}x + 13 = 0 \]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):

Here, \( a = 1, b = -\frac{53}{2}, c = 13 \):

\[ x = \frac{\frac{53}{2} \pm \sqrt{\left(-\frac{53}{2}\right)^2 - 4 \cdot 1 \cdot 13}}{2 \cdot 1} \]
\[ x = \frac{\frac{53}{2} \pm \sqrt{\frac{2809}{4} - 52}}{2} \]
\[ x = \frac{\frac{53}{2} \pm \sqrt{\frac{2809 - 208}{4}}}{2} \]
\[ x = \frac{\frac{53}{2} \pm \sqrt{\frac{2601}{4}}}{2} \]
\[ x = \frac{\frac{53}{2} \pm \frac{51}{2}}{2} \]

Calculating the two possible values:

1. \( x = \frac{104}{4} = 26 \)
2. \( x = \frac{2}{4} = \frac{1}{2} \)

### Final Zeros
The zeros of the cubic polynomial \( x^3 - \frac{53}{2}x^2 + 13x \) are:

\[ x = 0, \quad x = \frac{1}{2}, \quad x = 26 \]